Similarly to Lemma 2 in Cai and Liu [1]}, we could have
\(\Pr \left(\max _{i,j} | \bar{X}_i^{(j)\ast }| \ge C_2\sqrt{\frac{\ln p}{n_{ij}^\ast }}\right) = O\left(p^{-M}\right),\) 
\(\Pr \left(\max _{ij} \left|\frac{1}{n_{ij}^\ast }\sum _{k=1}^nS_{ik}S_{jk} X_{ik}^2 X_{jk}\bar{X}_j^\ast \right| \ge C_5\sqrt{\frac{\ln p}{n_{ij}^\ast }}\right).\) 
